A Krein-Like Formula for Singular Perturbations of Self-Adjoint Operators and Applications

Abstract

Given a self-adjoint operator $A:D(A)\subseteq\calH\to\calH$ and a continuous linear operator $τ:D(A)\to\X$ with Range$ τ'\cap\calH' ={0}$, $\X$ a Banach space, we explicitly construct a family $A^τ_Θ$ of self-adjoint operators such that any $A^τ_Θ$ coincides with the original $A$ on the kernel of $τ$. Such a family is obtained by giving a Kre\uın-like formula where the role of the deficiency spaces is played by the dual pair $(\X,\X')$; the parameter $Θ$ belongs to the space of symmetric operators from $\X'$ to $\X$. When $\X=\C$ one recovers the ``$\calH_{-2}$ -construction'' of Kiselev and Simon and so, to some extent, our results can be regarded as an extension of it to the infinite rank case. Considering the situation in which $\calH=L^2(\RE^n)$ and $τ$ is the trace (restriction) operator along some null subset, we give various applications to singular perturbations of non necessarily elliptic pseudo-differential operators, thus unifying and extending previously known results.